Problem: Factor the quadratic expression completely. $-3x^2+17x-20=$
Solution: Since the terms in the expression do not share a common monomial factor and the coefficient on the leading $x^2$ term is not $1$, let's factor by grouping. The expression ${-3}x^2{+17}x{-20}$ is in the form ${A}x^2+{B}x+{C}$. First, we need to find two integers ${a}$ and ${b}$ such that: $\begin{cases} &{a}+{b}={B}={17} \\\\ &{ab}={A}{C}= ({-3})({-20})=60 \end{cases}$ We find that ${a}={12}$ and ${b}={5}$ satisfy these conditions, since ${12}+{5}={17}$ and $({12})({5})=60$. Next, we can use these values to rewrite the $x$ -term and factor by grouping. $\begin{aligned} -3x^2+17x-20&=-3x^2+{12}x+{5}x-20 \\\\ &=-3x(x-4)+5(x-4) \\\\ &=(x-4)(-3x+5) \end{aligned}$ In conclusion, $-3x^2+17x-20=(x-4)(-3x+5)$